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J. Korean Math. Soc. 2018; 55(1): 185-210

Online first article November 21, 2017      Printed January 1, 2018

https://doi.org/10.4134/JKMS.j170110

Copyright © The Korean Mathematical Society.

Special values and integral representations for the Hurwitz-type Euler zeta functions

Su Hu, Daeyeoul Kim, Min-Soo Kim

South China University of Technology, Chonbuk National University, Kyungnam University

Abstract

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The Hurwitz-type Euler zeta function is defined as a deformation of the Hurwitz zeta function: \begin{equation*} \zeta_E(s,x)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+x)^s}. \end{equation*} In this paper, by using the method of Fourier expansions, we shall evaluate several integrals with integrands involving Hurwitz-type Euler zeta functions $\zeta_E(s,x)$. Furthermore, the relations between the values of a class of the Hurwitz-type (or Lerch-type) Euler zeta functions at rational arguments have also been given.

Keywords: Hurwitz zeta functions, Euler polynomials, integrals, Fourier series

MSC numbers: 33B15, 33E20, 11M35, 11B68, 11S80

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